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makePfisterForm -- the Grothendieck-Witt class of a Pfister form

Description

Given a sequence of elements $a_1,\ldots,a_n \in k$ we can form the Pfister form $\langle\langle a_1,\ldots,a_n\rangle\rangle$ defined to be the rank $2^n$ form defined as the product $\langle 1, -a_1\rangle \otimes \cdots \otimes \langle 1, -a_n \rangle$.

i1 : makePfisterForm(QQ, (2,6))

o1 = | 1 0  0  0  |
     | 0 -6 0  0  |
     | 0 0  -2 0  |
     | 0 0  0  12 |

o1 : GrothendieckWittClass

Inputting a ring element, an integer, or a rational instead of a sequence will produce a one-fold Pfister form instead. For instance:

i2 : makePfisterForm(GF(13), -2/3)

o2 = | 1 0 |
     | 0 5 |

o2 : GrothendieckWittClass
i3 : makePfisterForm(CC, 3)

o3 = | 1 0  |
     | 0 -3 |

o3 : GrothendieckWittClass

Ways to use makePfisterForm:

  • makePfisterForm(InexactFieldFamily,QQ)
  • makePfisterForm(InexactFieldFamily,RingElement)
  • makePfisterForm(InexactFieldFamily,Sequence)
  • makePfisterForm(InexactFieldFamily,ZZ)
  • makePfisterForm(Ring,QQ)
  • makePfisterForm(Ring,RingElement)
  • makePfisterForm(Ring,Sequence)
  • makePfisterForm(Ring,ZZ)

For the programmer

The object makePfisterForm is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/BuildingFormsDoc.m2:50:0.